1.9 Perpendicular unit vector* Find a unit vector perpendicular to A = (î + j- k) and B = j- 3k). (2î +

Q: 1) A ball is tossed directly south from an initial height of 20 meters, with an initial velocity of…

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Q: Two Electric field vectors E1 and E2 are perpendicular to each other; obtain its base  vectors.

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Q: 4.) Correct of K vector representation for of terms B the velatie. positive normal and tangential…

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Q: (b) Evaluate the velocity vector and acceleration vector of the object at the given point. %3D %3D…

A: As you mentioned part b is solved

Q: Find the magnitude ||0|| of the vector ü = i- 3j. Round your answer to 2 places after the decimal…

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Q: Ps Fnd Parametric equations for the 3. lines through (2,3,0) Perpendliculerr to the Vector uiitzj+3k…

A: First take cross product ie normal vector And then use formula

Q: 5Let MH)=Hit2costj + (2shnt-3k a) Find unit tingent vector TH) of rtH)

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Q: A force F = 2i +j - 3k is applied to a spacecraft with velocity vector v = 3i - j. Express F as a…

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Q: Express the vectors in Exercises 9–10 in terms of their lengths and directions. 9. Velocity vector v…

A: Since you have submitted two questions, we'll answer the first question. For the second question…

Q: 4. Find a unit vector in the same direction as the vector 8i- j+ 4k.

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Q: Find a normal vector at point P. xy2 - 4x2 + y + 8x = 7, P(4, 3) Find a tangent vector at point P.

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Q: Two vectors of magnitudes B= 5 and C = 3 are lying on the xy plane as shown in the figure. What is…

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Q: QI. If R = ti + 3 sin t j + 3 cos tk is the position vector of a particle in space. %3D Determine…

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Q: 1 u (-y°i +x rmal vector to

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Q: The position vector of a moving particle in space at time t is given by the vector function r (t) =…

A: position vector of particle in space r→(t)=12(5-t)2i→+(2t-1)j→+43(5-t)32k→

Q: 6. () Let r(ø, 0) = a sin ø cos đi+a sin ø sin 0j+a cos øk be the vector equa- tion of a sphere of…

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Q: ) A three dimensional motion of an object is given by the vector function r(t) = 4 cost i+ 4 sin t j…

A: The motion of the object is sketched as:

Q: which vector is NOT orthogonal to (cos θ, sin θ, 1)? a) (-sin θ, cos θ, 0) b) (-cos θ, -sin θ, 1)…

A: To find which vector is not orthogonal to (cosθ,sinθ,1)

Q: A particle moves so that its position vector r at timet is r = a cos wt + bsin wt, where w is a…

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Q: An eagle rises in a helical path given by the vector function r(t) =〈250cost, 250sint, 100t〉 where r…

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Q: Find a vector parametrization r(t) for the line with the given description. Passes through (-2, 0,…

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Q: Q/ find all unit vectors parallel to the yz plane and perpendicular to the vector V= 5i - j + 2k ?

A: Given,   A vector V= 5i - j + 2k.

Q: Q1A)The vectors A = 3i+j+ 4k, B = 2i - 3j-k, find vector normal to A and In plane of A and, B.…

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Q: 2 Let u = = 1 (i) Find the angle between u and v. (ii) Find a vector w of length 2 in the direction…

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Q: 6. Find the work done by the force F = 31 – 5j in moving an object from the point (2, 2) to the…

A: Given that the force F=3i-5j moving from the point (2,2) to the point (7,-13). We need to find the…

Q: Two vectors of magnitudes A=3 and B=3v2 are lying on the xy plane as shown in the figure. What is…

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Q: Find the angle 0 between the vectors. 5T u-(cos , sin S), v-(cos, sin 4) = 4 4 3

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Q: Q1: A force of 9 newtons is applied in the direction of the vector from P,(1,0,1) to P2(3,2,0).…

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Q: 2.10 Express the following vectors in rectangular coordinates: 2z az (a) A = p sin o a, + p cos o as…

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Q: Show that the vectors a = 3i+ 4j and b = 4i - 3j are perpendicular. A smooth sphere A of mass 2m…

A: Consider the given: a→=3i^+4j^b→=4i^-3j^ mass M of sphere A=2m velocity of sphere A of mass…

Q: 7. For vectors a and b in R? prove Lagrange's identity: ||a x b|? = ||a|| ||b||? – (a · b)?

A: for vectors a and b in ℝ2 prove -lagrange's identity a×b2=a2b2-a·b2

Q: 2. If A is the position vector of the point p-(x,y,z) and A=A. And ifr is the radius of the point in…

A: We apply the cross product to find the given function and simplification needed.

Q: ☺ Find the unit fenpent and unit normal vectors to the space )curve (circulur helix) determined by…

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Q: Express in cylindrical components: (a) the vector from C(3, 2, –7) to D(-1, –4,2); (b) a unit vector…

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Q: Find all unit vectors parallel to the JZ plane and perpendicular to the vector V = 5i -j + 2K %3D

A: A vector is parallel to yz-plane and perpendicular to the vector v, we have to find the unit…

Q: Q5: Plot the function y= 2 tan(t)-cos(mt) labeling the x and y axes. Where t is a vector containing…

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Q: Find the unit tangent vector T, the unit normal vector N, and the binormal vector B for curve r at…

A: Hey, since there are multiple subparts posted, we will answer first 3 questions. If you want any…

Q: F

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Q: Q1) Find equation of plane normal to vector 3i + 2j – k and pass through point (1,1,5).

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Q: A force F = 2i + j - 3k is applied to a spacecraft with velocity vector v = 3i - j . Express F as a…

A: We have given the force F = 2i + j - 3k with velocity vector v = 3i – j.

Q: Find an expression for a unit vector normal to the surface x = 8 cos (0) sin (4),y = 5 sin (0) sin…

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Q: aj Jai aj lai ow consider the circle in a general position where the base vectors i & j are replaced…

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Q: Find a vector parametrization for the line with the given description. Perpendicular to the…

A: Given points are (0,0,8). The equation of yz-plane is given by ⇒x=0 Therefore, the normal vector is…

Q: 1. A certain velocity vector V has an X-component, V,, of 12 m/s, and a Y-component, V, of 9 m/s.…

A: To find the magnitude of the velocity vector V.

Q: 9. Find the normal unit vector n to the surface 62 + 2y²+ 2² = 225 at the point P: (5,5, 5).

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Q: Find The vector novmal to The vecton A=2i-3; Ans/ N= 3i+2j Note The normal vector to The vecton A=…

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Q: A force of 15 Newton parallel to the vector V=3i+4j is applied to a point that moves on a line from…

A: We have to find work done.

Q: 2) a) Find the unit vectors that are parallel to the tangent line to the curve y = 2sinx at the…

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Q: Let vector u = and vector v . Find the magnitude of 3u-4v. Round to one decimal place.

A: Let vector u= <3, 4> and vector v = <6, 7> then find the magnitude of vector 3u-4v.

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: